34.7. SOME IMBEDDING THEOREMS 1215

Corollary 34.6.5 If Bu(0) = 0 for u ∈ X , then ⟨Bu,u⟩(0) = 0. The converse is also true.An analogous result will hold with 0 replaced with T .

Proof: Let un → u in X with un (t) = 0 for all t close enough to 0. For t off a set ofmeasure zero consisting of the union of sets of measure zero corresponding to un and u,

⟨Bun,un⟩(t) = ⟨B(t)un (t) ,un (t)⟩ ,⟨Bu,u⟩(t) = ⟨B(t)u(t) ,u(t)⟩ ,

⟨B(u−un) ,u⟩(t) = ⟨B(t)(u(t)−un (t)) ,u(t)⟩⟨Bun,u−un⟩(t) = ⟨B(t)un (t) ,u(t)−un (t)⟩

Then, considering such t,

⟨B(t)u(t) ,u(t)⟩−⟨B(t)un (t) ,un (t)⟩ = ⟨B(t)(u(t)−un (t)) ,u(t)⟩+⟨B(t)un (t) ,u(t)−un (t)⟩

Hence from Theorem 34.6.4,

|⟨B(t)u(t) ,u(t)⟩−⟨B(t)un (t) ,un (t)⟩| ≤C ||u−un||X (||u||X + ||un||X )

Thus if n is sufficiently large,

|⟨B(t)u(t) ,u(t)⟩−⟨B(t)un (t) ,un (t)⟩|< ε

So let n be fixed and this large and now let tk→ 0 to obtain ⟨B(tk)un (tk) ,un (tk)⟩= 0 for klarge enough. Hence

⟨Bu,u⟩(0) = limk→∞

⟨B(tk)u(tk) ,u(tk)⟩< ε

Since ε is arbitrary, ⟨Bu,u⟩(0) = 0.Next suppose ⟨Bu,u⟩(0) = 0. Then letting v ∈ X , with v smooth,

⟨Bu(0) ,v(0)⟩= ⟨Bu,v⟩(0) = ⟨Bu,u⟩1/2 (0)⟨Bv,v⟩1/2 (0) = 0

and it follows that Bu(0) = 0.

34.7 Some Imbedding TheoremsThe next theorem is very useful in getting estimates in partial differential equations. It iscalled Erling’s lemma.

Definition 34.7.1 Let E,W be Banach spaces such that E ⊆Wand the injection map fromE into W is continuous. The injection map is said to be compact if every bounded set in Ehas compact closure in W. In other words, if a sequence is bounded in E it has a convergentsubsequence converging in W. This is also referred to by saying that bounded sets in E areprecompact in W.